When should I use an alert and when the chat?

They are two complementary modes:

• Chat (Ask): when you have a specific question today and want an answer right away. Ad-hoc, on-demand research.
• Alerts (Anticipate): when you want to make sure you don't miss anything on a topic in the future. 24/7 monitoring without opening the tool.

Most professional users end up combining both: they use the chat to understand a topic in depth and, at the end, create an alert to stay up to date with future developments on that topic.